the one-to-one correspondence thus established between cor 
responding points of the two ranges is called a projec- 
tivity. 
To obtain an analytic property of a projectivity, we apply 
the sine proportion to two triangles in Fig. 1 and get 
AC _sin AVC BC _sin BVC 
AV sin ACV’ BV~sin ACV 
From these and the formulas with D in place of C, we get 
AC AVsinAVC AD_AV sinAVD 
BC BV' sin BVC BD~BV sin BVD‘ 
Hence, by division 
AC . AD _ sin A VC # sin A VD 
lBC~lBD^sin BVC^sin BVD' 
The left member is denoted by {ABCD) and is called the 
cross-ratio of the four points taken in this order. Since the 
right member depends only on the angles at V, it follows that 
{ABCD) = {AiBiCiD l ), 
if A h . . . , D i are the intersections of the four rays by a 
second transversal. Hence if two ranges are projective, the 
cross-ratio of any four points of one range equals the cross 
ratio of the corresponding points of the other range. 
Let each point of the line AB be determined by its dis 
tance and direction from a fixed initial point of the line; let 
a be the resulting coordinate of A, and b, c, x those of B, 
C, D, respectively. Similarly, let A', B', C', D' have the 
coordinates a', b', c', x', referred to a fixed initial point on 
their line. Then 
Hence 
x' — b' _ u x — b j _c 
» # i fc 
■a c —a 
x —a 
x — a 
V, not on their 
This pencil is 
. . ., said to be 
reject the points 
pencil and cut it 
)f points A', B' 
5 e A, B, C, . . . 
ir of projections 
iven range, and 
he points on a 
its. Project the 
and combinations 
re transformation 
from the origin 
retching perpen- 
tio 2 : 1. 
transformations, 
variants the very 
an interpretation 
ie ordinary needs